I would have killed for content like this back when I was getting my Physics degree. The diagrams are so beautiful and go straight to the heart of the key vector calculus concepts needed for E&M.
I remember struggling through Jackson[1] as a rite of passage, but there's no reason future generations should have to suffer as we did. This is what the web was meant to be.
Thanks a lot man - I'm really happy to have this kind of feedback. The reason I wrote this is because I found most of the modern explanations lacking in intuition behind the equations - along with also not explaining what the actual equations meant. If you found this useful please share and subscribe - I'm also trying to provide intuitive guides to other concepts (Schrodinger's equation, Black Holes, Quantum Mechanics, other complex topics) and eventually I'm hoping to write books on some of these topics which present math and physics in a much more clear and intuitive manner. Math shouldn't be hard to grasp. At the very bottom level it's very simple but presenting it in a clear and intuitive manner I will admit is very hard. Also full credit to a lot of the material as well goes to Grant Sanderson (3Blue1Brown) and most of the diagrams there were generated using Vexlio which I also highly promote: https://vexlio.com/
Also check out the YouTube videos of eigenchris, especially his series on tensor calculus and relativity. Probably the clearest explanations I’ve seen on these subjects.
I wanted to command you on your excellent work! I am curious how easy is to use Vexlio, is a steep curve? And any favorite books you want to share, I'm going on holidays soon, and haven't planned much for reading yet.
This is really excellent. I particularly like the outline of div and curl, the dot product and the cross product, and the connections drawn between the differential an integral forms. Thanks.
Thanks @photon_lines! In your temperature diagram, you mention that every point will take the average of the neighboring points. However, the equation is not a constraint on the temperature but on the "change of the slope (or gradient) of the temperature". The bigger the slope (in space), the faster (in time) the temperature changes at that point!
Speaking of 3Blue1Brown - would you be interested in presenting this same material in a video? 3Blue1Brown has regular showcases for this type of material. I will read your article when I get a moment but after a hard day's of screen time I sometimes like to kick back to a video.
I agree with the parent comment that the article was quite good and useful, although I do have a nit to pick with the section on unification of the electric and magnetic fields. I think needs to look at an additional scenario.
That section looks at three scenarios:
1. An electrically neutral straight wire with an electron current and a test charge near the wire moving in parallel to it at the same velocity as the electrons in the electron current, observed from an observer stationary with respect to the positive charges in the wire analyzed without taking into account relativity.
The analysis shows that there is no electrostatic force on the test charge because the wire is electrically neutral, but there is a magnetic force because the test charge is moving in the magnetic field caused by the electron current.
(Nit within a nit: the drawing for this shows the positive and negative charges in the wire separated with the positive charges quite a bit closer to the test charge. That would result in an electric field from the wire that would attract the test charge. Maybe insert a short note saying that the positive and negative charges in the wire are actually mixed together so that their electric fields cancel outside the wire?)
2. Same as #1 except the observer is stationary with respect to the test charge.
The observer now sees no electron current in the wire, but does see a current from the positive charges. But the magnetic field from that positive current should not exert a force on the test charge because magnetic fields only affect moving charges and the test charge is not moving in the observer's frame.
3. The Lorentz contraction is introduced, and #2 is re-analyzed taking that into account. That Lorentz contraction applied to the positive current manifests to the observer as an increased density of positive charges. There wire now appears to the observer to no longer be electrically neutral. It has a net positive charge and the resulting electric fields attracts the electron to the wire.
What's missing is circling back and looking at scenario #1 again but including the Lorentz contraction. In scenario #1 the observer sees the negative charges moving, so should see increased negative charge density due to the Lorentz contraction, and the wire should appear to them to have a net negative charge, which would try to repel the test charge.
#1 with Lorentz included then is a fight between the magnetic attraction and the electrostatic repulsion.
Assuming objective reality and so requiring the test charge to actually feel the same force no matter who is observing we can infer that if the electrostatic force toward the wire in #3 is F then the magnetic force toward the wire in #1 must be 2F, which when opposed by the -F electrostatic force from the Lorentz contraction of the negative charges in the wire gives a net force toward the wire of F.
1. While the posted guide is excellently written, it's not particularly novel. I was taught EM in a very similar fashion. Diagrams similar to those in the guide were drawn on the board by my professors.
2. Jackson is a graduate EM text. It is mathematically difficult, because when you read it, you should have been familiar with EM and all this conceptual underpinning for at least 3-4 years. The goal of Jackson is to solve the equations for scenarios that undergrads would find challenging. What did you study in your undergrad?
'What did you study in your undergrad?' - Computer Science. I study applied math and physics in my spare time - I'm currently teaching myself quantum field theory and other topics. For the most part - they're incomprehensible to an average person which is why I'm so passionate about doing what I'm doing - all of this stuff is extremely simple underneath but we humans find ways to make it complicated. Why not untangle that complexity and simply explain things in a clear and intuitive manner? Also - your comments on your undergraduate ease of grasping Maxwell's equations usually don't apply to everyone. Many professors don't sketch out what they mean and many books don't go through the fundamentals that students need in order to grasp what they mean. This guide is supposed to give someone a good background on 1) what they need to understand in order to grasp the equations and 2) what the equations actually mean in clear human language. Hopefully this helps - I also haven't had a chance to read Jackson but he's been mentioned so many times that right now I'll make a note to actually read the book and see how well he explains the concepts and see if I can maybe find other ways of making things simpler.
Re #2: Jackson was the standard text for undergrads like me doing a Mathematics degree. It was a late second year or early third (final) year text if I recall rightly. This was 1992, so I'm still amazed to read that its still a commonly used text.
Fwiw, other standard texts used in Durham (UK) back then were Spivak on Calculus, Goldstein on mechanics, and for the mathematical physics kids, landau and lifschitz on mechanics and electromagnetism, and (an absolute doorstop) Misner, Wheeler and Thorne on Gravitation (relativity).
People often make comments like this, forgetting that want they are marvelling at was actually in the book they read or class they took the first, and then actual different is that they forgot, or they've had more time to stew on the material so it feels more familiar the second time through. Hence the adage that the best book on the subject is whatever book you read second. It seems so much more intuitive the second time through.
We used the John Kraus book on Electromagnetics for the dynamic fields course. This was preceded by a course on static fields. That course’s final had the shortest test statement I had ever encountered: “Derive Maxwell’s Equations”. I found the Kraus book satisfactory.
At least it wasn't Halliday and Resnick. It's been 35 years since my BSME at Purdue, and I can still remember their names. God I hated those textbooks. If someone tells me that this Jackson book was worse, I won't believe it.
It really is a shame that in the 20th Century, the "best" math and science books were judged not for their educational power, but for how difficult and impressive they were to fellow professionals. It seems as though the professors were afraid that they'd lose their lecturer jobs if the books were too educational on their own.
Subject matter experts are not experts in pedagogy. Because pedagogy is a seperate subject entirely. And teaching is not about getting people to say aha. Thats just performance or entertainment. Seen everywhere these days thanks to the Attention Econnomy. You can gets ahas out of people playing great music. But dont equate that with getting people to play great music. Cuz that requires getting people to do lot of mundane mindless work for long long periods of time.
I don’t think this is true at all… Jackson is a good book not because it’s an easy introduction to EM but because it exposes you to more complex problems than would typically be looked at in undergraduate courses. There’s clearly a place for advanced texts for this reason.
Yeah reading this a couple decades out from my undergrad physics classes, my thought was "I remember learning all of this very painstakingly over multiple years and multiple different classes".
But also, I'm not sure I would have grokked much in this article without having taken those classes already, with the benefit of lectures and graded homework and group study sessions and TAs answering questions and all that...
The basic ideas, the pictures, the diagrams, etc, found here, typically show up in enough books if you look for them that I don't feel that this was the main limiting factor in my physics education. The difficulty of Jackson (which doesn't show up until grad school for most students) is in the problem sets, not the ideas behind the equations (which most students have a had at least two courses in already).
I don't believe that having a more 'intuitive' idea of the equations really helps all that much, as the intuition needed for solving the problems isn't really physical, but mathematical. Which integrals are solvable, which order of integration will make this tractable, do I need to use properties of Bessel functions here, etc.
We can argue whether getting good at this sort of thing is actually useful for physicists, but I wouldn't know. Very few of us ended up becoming researchers in the field.
“Maxwell’s theory only becomes simple and elegant once we start to think of the fields (mathematical functions) as being primary and the electromagnetic stresses and mechanical forces as being a consequence of such fields, and not vice-versa.”
I would have assumed Rupert Sheldrake would be completely taboo to most on HN, so I'm glad to see your comment voted highly! His telepathy "pseudo" scientific work is what he often leads into with any discussions of fields, and this was my favorite talk he gave: https://www.youtube.com/watch?v=GS1Drlyr37Q
This is incredibly well explained. Everything is simple, yet it is packed with so much details that memorising this and understanding this cannot be done without effort and focus. This whole stuff is fascinating when explained in such a way that it makes sense. I fought with this during my 2nd year of engineering studies, but did certainly not understand half of it at the time. With that explanation, I would have enjoyed studying the subject so much more. I guess I was not smart enough to understand my textbook and all the consequences of the formulas, so that I was unable to be fascinated by the subject.
I love articles like this, but they all make the same mistake of starting with vector algebra instead of geometric algebra. In 3D space, vector algebra works, but it falls flat on its face in both 2D and 4D scenarios. It's intuitive until it is completely broken.
I would love to see the same style of article, but using bivectors and the like where appropriate, such that the whole thing generalises neatly to 4D space-time, not just 3D space.
I will probably get downvoted for pointing this out, but the reality is that the geometric algebra approach to E&M, while interesting for its own reasons, will not replace the formalism based on Gibbs's vector calculus. One reason is simply that vector calculus is pretty intuitive and easy to learn. The major reason, however, is that the vector calculus approach is totally entrenched in the worlds of engineering and physics. After 100 years, nobody actually practicing those disciplines will make the notation change just so they can replace the 4 Maxwell's equations with one geometric algebra equation.
Also, Gibbs's vector calculus is used in fluid dynamics and other engineering disciplines, and as far as I know, nobody it touting the advantages of geometric algebra to folks working in fluid dynamics. I can be pretty sure that some HN reader will show me I am wrong about this by pointing out one lonely researcher who has found a way to express the Navier-Stokes equations using the geometric product ... but so what? ... My main point is that traditional vector calculus is a language everybody knows how to speak, geometric algebra is just another way to say the same things, so why would anybody change?
The metric system seems like a similar analog to geometric algebra vs vector calculus. You are saying the same thing but the language you are using is much more internally consistent.
Adoption has been bumpy given the US resistance but I think in the long run it (or something even more consistent) will win out. Similarly I think geometric algebra will be adopted. Maybe not in our lifetimes but eventually.
I actually took a look at doing this, but most human minds aren't tuned to 4D space-time, so if you have ideas on presenting this sort of thing to most people let me know and I'll be more than happy to modify my approach!!
tl;dr: GA's geometric product is a mixed-grade differential form, which is quite weird. Why not just think in terms of differential forms? Maxwell's equations are so sweetly summarized as dF=0 and d*F = J.
Just to give a brief answer to those reasonable criticisms:
The mixed-grade already exists in complex numbers (it is very useful there, and even more so in geometric algebra).
Differential forms are included in geometric algebra (the exterior/outer products are isomorphic). Turns out, combining that product with the inner product gives you an invertible product (as Clifford found out). That by itself already is a huge advantage.
Finally, Maxwell's equations are sweetly summarized in differential forms, but even more in geometric algebra: dF = J . Not only it is just one equation instead of two, but in addition the "d" (or "nabla") is directly invertible thanks to the geometric product (which differential forms lack and then have to use more indirect methods, including the Hodge dual).
By the way, I'm very partial to geometric algebra, but wouldn't say it is an "error" not to use it! Maybe just a big missed opportunity :)
Every vector calculus instructor should teach their students the intuitive (by which I mean visual/physical) meaning of grad, div, and curl (and the intuition behind results like Stokes's and Gauss's theorems). Even engineering students uninterested in proofs should be able to grok the intuition.
The textbook my Emag professor wrote himself made sure to avoid anything intuitive or visual, and was just a dense tome of text and equations with nothing else. He had a lot of trouble getting it published, but made sure to teach from it for decades. If you asked nicely, he'd give you a copy of the errata that he never fixed in the book. That class was essentially "vector calc for EEs" so it was my introduction to all these concepts, and I never intuitively understood them until much later.
You can make similar kinds of videos for all 3 of them. That video shows a divergence free field since number of particles aren't changing, I easily see that since I know the intuitive explanation for divergence, it is useful to have intuition for those things.
Gradient is just the equivalent of slope but for higher than 1 dimension.
Edit: Or no, that field has divergence, I'm dumb I didn't watch the start, many particles accumulate at a few points, that is due to divergence. Divergence is essentially areas that attracts or repels particles in that simulation.
> Virtually every force we experience in everyday life (with the exception of gravity) is electromagnetic in origin. [...] It wasn’t until the arrival of Oliver Heaviside, who reformulated and simplified the equations [...]
Maxwell's original equations connected light and electricity. Maxwell's original 20 equations had 20 unknowns, using 'quaternion-based notation', which no one understood.
Heaviside restated Maxwell's 20 equations into 4 equations using vector calculus. The restatements helped with simplification, but I believe it wasn't without cost.
There's a lot that's still unexplained in our modern world, especially with regards to individual humans' experiences. I got a window on these as a taxi driver, where I was sent people who helped me figure out things I'd been wondering about.
There ought to be a link between electromagnetism and gravity, we just haven't figured it out yet. This wikipedia article was cited by Bing CoPilot in response to my query. It's above my pay grade, maybe one of you can translate it for me: https://en.wikipedia.org/wiki/Gravitoelectromagnetism
Here is a set of lecture slides on the changing form of Maxwell's equations including the component form (which was apparently Maxwell's very first version), the quaternion form which came second and then Heaviside's version[1]
Fun fact about Heaviside (that noone asked for) - he's also the guy who invented the "cover up" method of doing partial fraction decomposition quickly.[2]
I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful. He certainly got rid of the quaternions but that seems like a minor difference because Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part.
The major difference I found was that Maxwell was expressing things in terms of the scalar and vector potential (which is what you have to do in QED) whereas Heaviside got rid of that and just had an electric and magnetic field instead. I found that you need 7 of Maxwell's equations to derive the 4 Heaviside(?) equations.
If you actually wanted to embrace quaternions you could write the famous 4 equations as just two (using natural units):
> Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part.
Did Maxwell actually use quaternions? If I recall correctly, at least in A Treatise on Electricity and Magnetism, quaternions were not actually used. Instead, he did most things in Cartesian coordinates, and all equations were applied to a vector's x, y, z components tediously. But many sources claimed Maxwell used quaternions, including quotes from Lord Kelvin. My reading on this part of history is limited, so my guess is that he did use them in personal research or in later papers. On the other hand, some other physicists of the same era used quaternions extensively, including applying them to Maxwell's electromagnetism, that is a sure fact...
Coincidentally, A Treatise on Electricity and Magnetism was written as an overview all electromagnetic phenomena as a whole, so it paid very little special attention to the generation and transmission of electromagnetic waves. Combining that with its difficult math, the book would puzzle physicists for another decade before they see the light from the book, and made it a rather curious period of history in electromagnetism.
> I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful.
In 1885, Heaviside published Electromagnetic Induction and Its Propagation in The Electrician, and formulated what he called the "Duplex Form" of Maxwell's equation. This was a long series of papers published in several months, and later republished in Electric Papers, Volume I. Basically, following his physical intuition, he felt that electric and magnetic fields should be symmetric and generate each other, and that should be directly highlighted in equations.
The logic of the paper went like the following.
First, he started with a definition of electric current [1]:
C = kE
D = cE / 4π
Γ = C + D
in which, E denotes electric force, C denotes conduction current, k denotes specific conductivity constant, D denotes displacement current, and c denotes dielectric constant. Finally, Γ denotes true electric current, which is the sum of the conduction and displacement terms.
Next, a definition of magnetic current [2]:
B = µH
G = Ḃ / 4π = µḢ / 4π
G' = gH + µḢ / 4π
H denotes magnetic force, B denotes magnetic induction, µ denotes permeability, G denotes magnetic current, Ḃ and Ḣ are derivatives of B and H (Newton's notation). Hypothetically, suppose that magnetic monopoles exist (Heaviside did so), G' would denote the "true magnetic current", with an extra conduction term gH, where g is a constant similar to k.
Then, he introduced the concepts of divergence and curl, and their physical significance [3]. After more discussion and derivation, he finally wrote [4]:
in which, e and H denote impressed electric and magnetic forces to take static fields into account. Finally, since magnetic monopoles don't exist, he made g = 0, but kept this term in the equations for symmetry and elegance. [0]
This is the core of Heaviside's Duplex Form of Maxwell's equations. one can clearly see the co-evolution of electric and magnetic fields, and is the precursor of the modern Maxwell's equations as we know today in its vector calculus formulation. As far as I know, his treatment of "physical" vectors as first-class objects is his original invention (independently invented by Gibbs as well), although the concepts themselves came from quaternions.
This is not a complete summary, as he continued his analysis in a series of publications.
A good book on this part of history is Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, by Paul J. Nahin.
[0] So the claim "Maxwell's equations need modifications if magnetic monopole has been discovered" is historically inaccurate, it should rather be, "be restored to Heaviside's original form."
Gravitoelectromagnetism doesn't actually have anything to do with electromagnetism except that certain formal features of the theory of general relativity correspond roughly to the mathematical structures we talk about in electromagnetism, albeit with the proviso that the symmetries underlying the two theories are different.
Quaternion was crucial and instrumental tool in Maxwell discovery and the formulation of the electromagnetics (EM) equations. When Terence Tao was asked how come nobody has proof of the Riemann hypothesis, arguably the hardest of the Math problems, and according to him this is because there is no appropriate tools available at the moment to proof it. I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis.
Unlike other waves for example sound waves, EM has a unique polarization property. In order to completely and correctly model EM based phenomena quaternion based formulation and representation is necessary. One of the reasons that almost all existing wireless modulation are not utilizing polarization is due to most of the microwave and wireless engineers are not familiar with quaternion. Ironically their biased attitude is not unlike early mathematicians and scientists that were very much opposed to complex number, and it turn out that almost all of the modern wireless modulation for example OFDM are utilizing complex number.
For the derivation of the Maxwell’s equations using geometric algebra involving quaternion please check these articles and they can be summarized the into one elegant equation [1][2].
[1] Maxwell’s eight equations as one quaternion equation:
Lay people seem to have this weird obsession with Quaternions and love to suggest that somehow theoretical physicists are missing something because they don't use them. But physicists are almost disgustingly familiar with SU(2) which is isomorphic to the quaternions and easier to work with and understand (quite obviously, in my opinion). It is hard to imagine, from my point of view, that a mere isomorphism stands between physicists and progress, especially given that physicists have long generalized _beyond_ SU(2) and the quaternions in their understanding of fundamental fields. Formulating an SU(3) gauge theory in terms of quaternions would at least be difficult and almost certainly be goofy, if not impossible.
As for "I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis" I'd love to understand your intuition here, because I just don't see it.
Some people are of the philosophical bent that our world is entirely random. But this doesn't commonly match our experience. For example, I often asked people how they met their significant other. Sometimes it was nothing special, some couples had quite a story.
I had the sense that I got certain passengers for more than just transportation. Some people were having a rotten day, and I was able to cheer them up. One lady had some time to kill before her bus' departure time, so we went to the 24 hour diner, ordered our own pies and compared notes. When we got to the bus station she said it was the best birthday she'd had in quite a long time.
The most important thing I learned in my taxi was about substance abuse. This HN poll didn't get any upvotes, but it references some of the diaries I never finished: https://news.ycombinator.com/item?id=39071316
Readit News
I remember struggling through Jackson[1] as a rite of passage, but there's no reason future generations should have to suffer as we did. This is what the web was meant to be.
[1]: https://en.wikipedia.org/wiki/Classical_Electrodynamics_(boo...
Deserves to be widely used to teach Maxwell's equations.
THANK YOU.
Deleted Comment
That section looks at three scenarios:
1. An electrically neutral straight wire with an electron current and a test charge near the wire moving in parallel to it at the same velocity as the electrons in the electron current, observed from an observer stationary with respect to the positive charges in the wire analyzed without taking into account relativity.
The analysis shows that there is no electrostatic force on the test charge because the wire is electrically neutral, but there is a magnetic force because the test charge is moving in the magnetic field caused by the electron current.
(Nit within a nit: the drawing for this shows the positive and negative charges in the wire separated with the positive charges quite a bit closer to the test charge. That would result in an electric field from the wire that would attract the test charge. Maybe insert a short note saying that the positive and negative charges in the wire are actually mixed together so that their electric fields cancel outside the wire?)
2. Same as #1 except the observer is stationary with respect to the test charge.
The observer now sees no electron current in the wire, but does see a current from the positive charges. But the magnetic field from that positive current should not exert a force on the test charge because magnetic fields only affect moving charges and the test charge is not moving in the observer's frame.
3. The Lorentz contraction is introduced, and #2 is re-analyzed taking that into account. That Lorentz contraction applied to the positive current manifests to the observer as an increased density of positive charges. There wire now appears to the observer to no longer be electrically neutral. It has a net positive charge and the resulting electric fields attracts the electron to the wire.
What's missing is circling back and looking at scenario #1 again but including the Lorentz contraction. In scenario #1 the observer sees the negative charges moving, so should see increased negative charge density due to the Lorentz contraction, and the wire should appear to them to have a net negative charge, which would try to repel the test charge.
#1 with Lorentz included then is a fight between the magnetic attraction and the electrostatic repulsion.
Assuming objective reality and so requiring the test charge to actually feel the same force no matter who is observing we can infer that if the electrostatic force toward the wire in #3 is F then the magnetic force toward the wire in #1 must be 2F, which when opposed by the -F electrostatic force from the Lorentz contraction of the negative charges in the wire gives a net force toward the wire of F.
1. While the posted guide is excellently written, it's not particularly novel. I was taught EM in a very similar fashion. Diagrams similar to those in the guide were drawn on the board by my professors.
2. Jackson is a graduate EM text. It is mathematically difficult, because when you read it, you should have been familiar with EM and all this conceptual underpinning for at least 3-4 years. The goal of Jackson is to solve the equations for scenarios that undergrads would find challenging. What did you study in your undergrad?
Fwiw, other standard texts used in Durham (UK) back then were Spivak on Calculus, Goldstein on mechanics, and for the mathematical physics kids, landau and lifschitz on mechanics and electromagnetism, and (an absolute doorstop) Misner, Wheeler and Thorne on Gravitation (relativity).
But also, I'm not sure I would have grokked much in this article without having taken those classes already, with the benefit of lectures and graded homework and group study sessions and TAs answering questions and all that...
I don't believe that having a more 'intuitive' idea of the equations really helps all that much, as the intuition needed for solving the problems isn't really physical, but mathematical. Which integrals are solvable, which order of integration will make this tractable, do I need to use properties of Bessel functions here, etc.
We can argue whether getting good at this sort of thing is actually useful for physicists, but I wouldn't know. Very few of us ended up becoming researchers in the field.
I later found out that you can squeeze even more beauty out of them by boiling them down even further using differential geometry.
http://virtualmath1.stanford.edu/~conrad/diffgeomPage/handou...
https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...
There is a lot of interesting discussion on whether fields are real, and the dialogue goes back centuries: https://youtu.be/j2oSyAfPzWg?si=BHRv8lodGhqZBtbl
I would love to see the same style of article, but using bivectors and the like where appropriate, such that the whole thing generalises neatly to 4D space-time, not just 3D space.
Also, Gibbs's vector calculus is used in fluid dynamics and other engineering disciplines, and as far as I know, nobody it touting the advantages of geometric algebra to folks working in fluid dynamics. I can be pretty sure that some HN reader will show me I am wrong about this by pointing out one lonely researcher who has found a way to express the Navier-Stokes equations using the geometric product ... but so what? ... My main point is that traditional vector calculus is a language everybody knows how to speak, geometric algebra is just another way to say the same things, so why would anybody change?
Adoption has been bumpy given the US resistance but I think in the long run it (or something even more consistent) will win out. Similarly I think geometric algebra will be adopted. Maybe not in our lifetimes but eventually.
https://vixra.org/pdf/1206.0021v1.pdf
tl;dr: GA's geometric product is a mixed-grade differential form, which is quite weird. Why not just think in terms of differential forms? Maxwell's equations are so sweetly summarized as dF=0 and d*F = J.
The mixed-grade already exists in complex numbers (it is very useful there, and even more so in geometric algebra).
Differential forms are included in geometric algebra (the exterior/outer products are isomorphic). Turns out, combining that product with the inner product gives you an invertible product (as Clifford found out). That by itself already is a huge advantage.
Finally, Maxwell's equations are sweetly summarized in differential forms, but even more in geometric algebra: dF = J . Not only it is just one equation instead of two, but in addition the "d" (or "nabla") is directly invertible thanks to the geometric product (which differential forms lack and then have to use more indirect methods, including the Hodge dual).
By the way, I'm very partial to geometric algebra, but wouldn't say it is an "error" not to use it! Maybe just a big missed opportunity :)
The brown Rudin.
https://www.youtube.com/watch?v=eEwZeY51mT0
You can make similar kinds of videos for all 3 of them. That video shows a divergence free field since number of particles aren't changing, I easily see that since I know the intuitive explanation for divergence, it is useful to have intuition for those things.
Gradient is just the equivalent of slope but for higher than 1 dimension.
Edit: Or no, that field has divergence, I'm dumb I didn't watch the start, many particles accumulate at a few points, that is due to divergence. Divergence is essentially areas that attracts or repels particles in that simulation.
Found the divergence video, in case it is hard to understand what I said above: https://www.youtube.com/watch?v=c0MR-vWiUPU
https://en.m.wikipedia.org/wiki/Del and the related articles on gradient (slope), divergence (flow across a boundary), and curl (circulation)
Maxwell's original equations connected light and electricity. Maxwell's original 20 equations had 20 unknowns, using 'quaternion-based notation', which no one understood.
Heaviside restated Maxwell's 20 equations into 4 equations using vector calculus. The restatements helped with simplification, but I believe it wasn't without cost.
There's a lot that's still unexplained in our modern world, especially with regards to individual humans' experiences. I got a window on these as a taxi driver, where I was sent people who helped me figure out things I'd been wondering about.
There ought to be a link between electromagnetism and gravity, we just haven't figured it out yet. This wikipedia article was cited by Bing CoPilot in response to my query. It's above my pay grade, maybe one of you can translate it for me: https://en.wikipedia.org/wiki/Gravitoelectromagnetism
Fun fact about Heaviside (that noone asked for) - he's also the guy who invented the "cover up" method of doing partial fraction decomposition quickly.[2]
[1] https://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl/Ma...
[2] https://math.mit.edu/~jorloff/suppnotes/suppnotes03/h.pdf
The major difference I found was that Maxwell was expressing things in terms of the scalar and vector potential (which is what you have to do in QED) whereas Heaviside got rid of that and just had an electric and magnetic field instead. I found that you need 7 of Maxwell's equations to derive the 4 Heaviside(?) equations.
If you actually wanted to embrace quaternions you could write the famous 4 equations as just two (using natural units):
∇E + dB/dt = -ρ
∇B - dE/dt = J
Did Maxwell actually use quaternions? If I recall correctly, at least in A Treatise on Electricity and Magnetism, quaternions were not actually used. Instead, he did most things in Cartesian coordinates, and all equations were applied to a vector's x, y, z components tediously. But many sources claimed Maxwell used quaternions, including quotes from Lord Kelvin. My reading on this part of history is limited, so my guess is that he did use them in personal research or in later papers. On the other hand, some other physicists of the same era used quaternions extensively, including applying them to Maxwell's electromagnetism, that is a sure fact...
Coincidentally, A Treatise on Electricity and Magnetism was written as an overview all electromagnetic phenomena as a whole, so it paid very little special attention to the generation and transmission of electromagnetic waves. Combining that with its difficult math, the book would puzzle physicists for another decade before they see the light from the book, and made it a rather curious period of history in electromagnetism.
> I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful.
In 1885, Heaviside published Electromagnetic Induction and Its Propagation in The Electrician, and formulated what he called the "Duplex Form" of Maxwell's equation. This was a long series of papers published in several months, and later republished in Electric Papers, Volume I. Basically, following his physical intuition, he felt that electric and magnetic fields should be symmetric and generate each other, and that should be directly highlighted in equations.
The logic of the paper went like the following.
First, he started with a definition of electric current [1]:
in which, E denotes electric force, C denotes conduction current, k denotes specific conductivity constant, D denotes displacement current, and c denotes dielectric constant. Finally, Γ denotes true electric current, which is the sum of the conduction and displacement terms.Next, a definition of magnetic current [2]:
H denotes magnetic force, B denotes magnetic induction, µ denotes permeability, G denotes magnetic current, Ḃ and Ḣ are derivatives of B and H (Newton's notation). Hypothetically, suppose that magnetic monopoles exist (Heaviside did so), G' would denote the "true magnetic current", with an extra conduction term gH, where g is a constant similar to k.Then, he introduced the concepts of divergence and curl, and their physical significance [3]. After more discussion and derivation, he finally wrote [4]:
in which, e and H denote impressed electric and magnetic forces to take static fields into account. Finally, since magnetic monopoles don't exist, he made g = 0, but kept this term in the equations for symmetry and elegance. [0]This is the core of Heaviside's Duplex Form of Maxwell's equations. one can clearly see the co-evolution of electric and magnetic fields, and is the precursor of the modern Maxwell's equations as we know today in its vector calculus formulation. As far as I know, his treatment of "physical" vectors as first-class objects is his original invention (independently invented by Gibbs as well), although the concepts themselves came from quaternions.
This is not a complete summary, as he continued his analysis in a series of publications.
A good book on this part of history is Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, by Paul J. Nahin.
[0] So the claim "Maxwell's equations need modifications if magnetic monopole has been discovered" is historically inaccurate, it should rather be, "be restored to Heaviside's original form."
[1] Electric Papers, Volume I, Page 429, https://archive.org/details/electricalpapers01heavuoft/page/...
[2] Page 441: https://archive.org/details/electricalpapers01heavuoft/page/...
[3] Page 443: https://archive.org/details/electricalpapers01heavuoft/page/...
[4] Page 449: https://archive.org/details/electricalpapers01heavuoft/page/...
Unlike other waves for example sound waves, EM has a unique polarization property. In order to completely and correctly model EM based phenomena quaternion based formulation and representation is necessary. One of the reasons that almost all existing wireless modulation are not utilizing polarization is due to most of the microwave and wireless engineers are not familiar with quaternion. Ironically their biased attitude is not unlike early mathematicians and scientists that were very much opposed to complex number, and it turn out that almost all of the modern wireless modulation for example OFDM are utilizing complex number.
For the derivation of the Maxwell’s equations using geometric algebra involving quaternion please check these articles and they can be summarized the into one elegant equation [1][2].
[1] Maxwell’s eight equations as one quaternion equation:
https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwe...
[2] A derivation of the quaternion Maxwell’s equations using geometric algebra:
https://peeterjoot.com/2018/03/05/a-derivation-of-the-quater...
As for "I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis" I'd love to understand your intuition here, because I just don't see it.
I'm curious now, would you indulge me? If its woo woo we can just pretend no one is reading :)
I had the sense that I got certain passengers for more than just transportation. Some people were having a rotten day, and I was able to cheer them up. One lady had some time to kill before her bus' departure time, so we went to the 24 hour diner, ordered our own pies and compared notes. When we got to the bus station she said it was the best birthday she'd had in quite a long time.
This was a semi-recent comment about the matching algorithm: https://news.ycombinator.com/item?id=34402081
The most important thing I learned in my taxi was about substance abuse. This HN poll didn't get any upvotes, but it references some of the diaries I never finished: https://news.ycombinator.com/item?id=39071316
Another comment: https://news.ycombinator.com/item?id=25238488
If you're so inclined, I'm curious if you've experience is also that our universe is more than random?
https://physics.stackexchange.com/questions/489291/how-did-e...